# Algebra switching – Field dividing an intermediate field

Consider an irreducible polynomial $$f (x) in F (x)$$ in a polynomial ring of a field $$F$$. Let $$L$$ to be the dividing field of $$f (x)$$ more than $$F$$. Suppose that $$L$$ is an extension of Galois on $$F$$. Let $$alpha in L$$ to be a root of $$f (x)$$. Consider an intermediate field $$L-K-F$$. Let $$g (x) in K (x)$$ to be the minimal polynomial of $$alpha$$. Let $$M$$ to be a subfield of $$L$$ which is the dividing field of $$g (x)$$. Have we $$M = L$$?

Assume that $$L = F (α)$$ is a normal separable extension of F. Then, it follows that for any intermediate field $$L – K – F$$, $$K (α) = L$$. I'm trying to generalize this. Of course if $$f (x)$$ in question is factored as $$g (x) h (x)$$. It may be that in the area of ​​the split of $$g (x)$$ more than $$K$$, $$h (x)$$ remains irreducible or at least not taken into account in linear polynomials. However, I do not find a counter-example.

If the answer to the question is negative, I have another question.

If the answer to the first question is negative, is there a plausible condition for having $$M = L$$?